
Penalty method: The penalty method supplements the weak form of boundary-fitted FEM with a penalty term that penalizes differences between the approximate solution and the prescribed Dirichlet data. This approach, which has been applied in the pioneering work on the finite cell method [1]}, is generally considered as the most straightforward technique to impose Dirichlet conditions on unfitted boundaries. The formulation omits the boundary terms that arise from the partial integration in the derivation of the weak form – in boundary-fitted FEM these terms drop out as the test functions vanish on Dirichlet boundaries – such that a modeling error is introduced which yields an inconsistent formulation. For appropriately selected penalty parameters the inconsistency can be acceptable [2]}, making the penalty method effective for a broad class of immersed problems with complex geometries, see, e.g., [3]}, [4]}. Nevertheless, the choice of an appropriate penalty parameter is challenging. A too small value does not adequately enforce the prescribed boundary conditions, while a too large value exacerbates the conditioning problems [5]} and can lead to
large, nonphysical, gradients on cut elements [6]}, [7]}, [8]}.

Nitsche's method: Nitsche's method [9]} can be considered as the consistent equivalent of the penalty method, as it retains the boundary gradients (sometimes referred to as the flux terms) in the weak formulation. Through appropriate scaling of the Nitsche (or penalty) parameter, a stable formulation is obtained, see, e.g., [10]}. Nitsche's method is a widely used technique for the weak imposition of boundary conditions in immersed finite element methods. An elegant aspect of Nitsche's method is that the parameters can be computed per element [11]}, avoiding potential difficulties in the selection of a single global Nitsche parameter, see, e.g., [12]}. The value of the Nitsche parameter should be inversely proportional to the thickness of the elementThis holds for second-order elliptic problems like the Poisson problem (REF ). With higher-order equations some parameters require higher scaling rates. [11]}, and can become arbitrarily large for small cut elements [8]}. This problem can be remedied by means of additional stabilization terms such as the ghost penalty, or by the aggregation of basis functions; see Section . Also, a nonsymmetric Nitsche method can be applied to avoid the need for stabilization [15]}, [16]}, [17]}, although this does affect the linear solver. Additionally, nonsymmetric Nitsche methods are not adjoint consistent, which means that these formulations result in suboptimal approximation properties in the \(L^2(\Omega )\) -norm [15]}.

Lagrange multiplier techniques: Dirichlet conditions on immersed boundaries can be enforced by supplementing the weak formulation with additional constraint terms, e.g., [19]}, [10]}. In contrast to the penalty method and Nitsche's method, in Lagrange multiplier techniques these constraint terms are associated with an auxiliary field variable which is defined over the unfitted boundary. This auxiliary field is referred to as the Lagrange multiplier field. Lagrange multiplier techniques result in a saddle point problem, which implies that the discrete Lagrange multiplier field needs to be selected in such a way that a stable system is obtained [21]}. Examples of Lagrange multiplier type techniques for immersed FEM are presented in [22]}, [23]}, [24]}, [25]}, [26]}. While an advantage of Lagrange multiplier techniques over Nitsche's method and the penalty method is that these do not require the selection of a parameter, the downsides are that additional degrees of freedom are introduced through the Lagrange multiplier field, and that a (inf-sup) stable discretization of the Lagrange multiplier field is generally non-trivial. Additionally, for many problems, the introduction of Lagrange multipliers changes the nature of the linear system from positive (semi-)definite to indefinite and breaks the diagonal dominance. This affects the applicability of iterative solvers (in particular this precludes the conjugate gradient method) and the factorization in sparse direct solvers.

Basis function redefinition: An alternative class of techniques to impose Dirichlet conditions on immersed boundaries is based on the idea to redefine the basis functions in such a way that the modified (non-vanishing) basis functions are interpolatory on the unfitted boundary. This enables traditional strong imposition of boundary conditions, as is standard in boundary-fitted finite element methods. Prominent examples of methods based on this concept are WEB-splines [27]}, [28]} and, more recently, i-splines [29]}. The main advantage of these techniques is that they do not require modifications to the weak formulation in comparison to boundary-fitted FEM, but the algorithms to redefine the basis functions constitute an additional non-trivial component in the implementation and analysis.

